Sobel on a Dark-to-Light Patch

The two Sobel kernels are

$G_x = \begin{bmatrix} -1 & 0 & +1 \ -2 & 0 & +2 \ -1 & 0 & +1 \end{bmatrix} \qquad G_y = \begin{bmatrix} -1 & -2 & -1 \ 0 & 0 & 0 \ +1 & +2 & +1 \end{bmatrix}$

Apply both to this 3x3 patch of a grayscale image, aligning each kernel weight with the pixel underneath it and summing the nine products.

row	col 1	col 2	col 3
top	10	10	200
middle	10	10	200
bottom	10	10	200

What are the two responses, and what do they say about the edge?

$G_x = 380$ , $G_y = 0$ : the kernel is separable, so only its centre row $(-2, 0, +2)$ contributes.

$G_x = 760$ , $G_y = 0$ : the intensity changes along the horizontal axis, so this is a vertical edge.

$G_x = 0$ , $G_y = 760$ : the kernel with the horizontal weight pattern is the one that fires on this edge.

$G_x = 760$ , $G_y = 760$ : any sharp intensity transition produces a large response in both kernels.